Plane waves and spacelike infinity

TL;DR

This paper investigates the structure of spacelike infinity in the causal boundary of homogeneous plane waves, revealing complex classifications of spacelike curves and the topological properties of their limits.

Contribution

It classifies the ways spacelike curves approach infinity in plane waves and explores the topological implications for the causal boundary.

Findings

Spacelike curves can approach infinity in multiple distinct ways.

The set of limits at spacelike infinity is two-dimensional.

Different topologies lead to either non-Hausdorff or incomplete causal boundaries.

Abstract

In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the null convergence condition consists of a single null curve. In Einstein-Hilbert gravity, this would include any homogeneous plane wave satisfying the weak null energy condition. For conformally flat plane waves such as the Penrose limit of $AdS_5 \times S^5$, all spacelike curves that reach infinity also end on this boundary and the completion is Hausdorff. However, the more generic case (including, e.g., the Penrose limits of $AdS_4 \times S^7$ and $AdS_7 \times S^4$) is more complicated. In one natural topology, not all spacelike curves have limit points in the causal completion, indicating the need to introduce additional points at `spacelike infinity'--the endpoints of spacelike curves. We classify the distinct ways in which spacelike curves can approach infinity, finding a {\it two}-dimensional set of distinct limits. The dimensionality of the set of points at spacelike infinity is not, however, fixed from this argument. In an alternative topology, the causal completion is already compact, but the completion is non-Hausdorff.